I live near a town with what I think is a similar intersection. The law here is that it is always the pedestrian's "turn" if they choose to enter a crosswalk and there is no traffic light specifically telling them otherwise. In more detail: if the pedestrian wishes to cross at an intersection with a traffic light, they have "right of way" (the legal term that corresponds to your use of "turn") when the light is green for the lane of travel parallel to their crosswalk; if there is no light, the pedestrian always has "right of way". For intersections with no light where pedestrians have difficulty crossing the road safety even in the crosswalk, we sometimes set up the warning lights situation. These lights serve the same function as the turn signals on a car: they broadcast a specific kind of intention so others using the road have an easy-to-see and harder-to-ignore signal, and so that everybody involved can feel comfortable that communication has been had.
Even normal lights, with signage that tell you that they make it your turn, are still having you assume the risk that the relationship between the crossing light and the crossing right is still present in the local traffic laws, they just aren’t so transparent about it. The car drivers aren’t taking their cues from the signage but from their beliefs about the lights.
I suppose if one believes the local authorities are drawing these fine distinctions, we might suppose that if the local laws did change, a philosophically-committed sign might be changed faster than a non-committed one, and therefore we might have slightly more confidence that one has the turn when crossing at those lights. It seems like a small effect.
I think the crux is right-of-way. Boats and ships have elaborate rules that always establish a right-of-way that can be clearly established after the fact, so all pilots and captains adhere their behavior to their expectations about the rules. The other thing about navigation on water is that in a close encounter the boat with the right-of-way is required to follow through so the other parties can predict what they can do. This is also not true on the road, leading to the phenomenon of drivers "politely" waving you to go out of turn.
The rules of the road aren't nearly as clear. (for instance, when they give the right of way to the car that arrived earlier at an intersection, that's not decidable after the fact).
The role of traffic lights is ensconced in the law, so pressing the button causes the lights to change, and the lights impact who has the right of way. It doesn't matter what the light thinks as long as the light changes.
Of course, I also remind myself at crosswalks that having the right-of-way is insufficient defense in an encounter between a pedestrian and a car.
Seeking to cross a road on the walk into downtown Lafayette, instead of the normal pedestrian crossing situation, we met a button with a sign, ‘Push button to turn on warning lights’. I wondered, if I pressed it, would it then be my turn to cross? Or would there just be some warning lights? What was the difference? Do traffic buttons normally do something other than change the lights? I clearly believe they do. They make it my turn. But they don’t send a wire into the ‘whose turn is it’ variable deep in the ought-sphere, so what exactly do they do?
I suspected that this button didn’t change whose turn it was, and it felt empty and devoid of some special potence of being a traffic button.
I liked to imagine that it was just a normal traffic button, but taking a more nihilistic view of its role. In which case, its nihilistic view seemed to have practical consequences! It wasn’t being as good a good traffic button while saying that it didn’t change whose turn it is. It genuinely fails to coordinate the traffic so well, because here am I unable to garner the ‘right’ to cross with confidence, and there are the drivers unsure what I’m doing. But shouldn’t a traffic button be able to do its job regardless of its philosophical commitments, or without pretending to have philosophical commitments it doesn’t have?
One might say that the thing going on is that it being ‘my turn’ is a fact about everyone’s expectations. For instance, if the drivers will expect me to cross, then it is ‘my turn’. (I’m tempted to say ‘if the drivers think it is my turn, then it is my turn’, but what are the drivers then thinking?) This doesn’t seem quite right, in that the drivers themselves are asking whether this light means that it is the pedestrian’s turn, and all of us seem to be asking something about the underlying truth, not about each other. Also, if I run erratically into the road, the drivers and I may both come to expect that I am going to cross, but it still isn’t my turn.
I fantasized that I had woken up in a new world which was just like the old world, but where everything was like the traffic light. I would phone the doctor’s office later to ask if it was ok to cancel my appointment this late, they would just say, ‘I’ll change what it says in this schedule’.
‘But is it ok?’
‘I will not record your cancellation.’
‘Should I pay you?’
‘I am not charging you’
‘But is that like a favor, or is it the policy? Have I wronged your medical practice? Do I owe you really? Tell me if I was bad!’
‘I erased your name from this box on my piece of paper.’
My tentative take is that turns are real, and we created them, and traffic buttons have genuine power over them, and if a traffic button doesn’t recognize their existence it is potentially at a real disadvantage, perhaps in a similar way to how a chair maker who doesn’t recognize chairs as a thing is at a disadvantage.
(To be clear, I expect philosophers have much better thought out views on this, and welcome people telling me what they are–this is me thinking aloud, not philosophical advice.)